FirstAtomicComponentBlockCount

The least l for which the first l blocks hold exactly 1 .. m.

✓ checked against that system · ! checked, with disagreements (hover a chip)FindStatSt000695via SetPartition✓↗

FirstAtomicComponentBlockCount(partition)The least l for which the first l blocks hold exactly 1 .. m.

Typed signatures

Type	Package	Applies
(set_partition) -> number	combinatorics	always

Domain: Set partition statistics

Details

• Defined over SetPartition as an expression in _x, evaluated by compute-engine — the definition IS the implementation.
• Takes a SetPartition and nothing else — it reads values against their positions, or walks the orbits, so it needs the bijection. Applying it to a bare list is a type error, not a wrong answer.

Definition

Epsil, as the implementation runs it.

on SetPartition, over _x

0 if Length(_x) < 1 else Min(Filter(
                                     Range(1, Length(_x)),
                                     Function(Max(Map(Function(
                                                                Last(_x[t]) if t <=
                                                                               l else 0,
                                                                t,
                                                              ), Range(
                                                                       1,
                                                                       Length(_x),
                                                                     ))) ==
                                              Sum(Map(Function(
                                                                Length(_x[t]) if t <= l else 0,
                                                                t,
                                                              ), Range(
                                                                       1,
                                                                       Length(_x),
                                                                     ))), l),
                                   ))

Examples

Basic #

the partition $\{1,3\} \mid \{2\}$

live ↗

In$\mathrm{FirstAtomicComponentBlockCount}({\operatorname{SetPartition}}([[1, 3], [2]]))$Out$2$

the partition $\{1\} \mid \{2,4\} \mid \{3\}$

live ↗

In$\mathrm{FirstAtomicComponentBlockCount}({\operatorname{SetPartition}}([[1], [2, 4], [3]]))$Out$1$

Properties #

FindStat's value for {{1,2}}

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SetPartition}}([[1, 2]]), \text{FirstAtomicComponentBlockCount})$Out$1$

Known value · FindStat St000695

FindStat's value for {{1,2,3}}

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SetPartition}}([[1, 2, 3]]), \text{FirstAtomicComponentBlockCount})$Out$1$

Known value · FindStat St000695

FindStat's value for {{1},{2}}

live ↗

In$\mathrm{CombinatorialStat}({\operatorname{SetPartition}}([[1], [2]]), \text{FirstAtomicComponentBlockCount})$Out$1$

Known value · FindStat St000695

About

Examples	5
Crosswalk	FindStat